Intuitively, by consistency of posteriors [1]}, as the number of arm allocations \(N_k \rightarrow \infty \) , \(k=0,1\) , the expected values of the posterior parameters distributions converge to the true mean rewards \(\mu _k\) , with \(k=0,1\) .
Furthermore, the posteriors' variance tends to zero. Thus, if an optimal arm exists, it will gradually and eventually reveal itself, and the probability of allocating that arm by drawing a value \(\tilde{\mu }_1 > \tilde{\mu }_0\) , according to TS, will tend to unity. As discussed in [2]}, if for any finite number of trials, the probability \(\pi _{t,k}\)  of allocating arm \(k\)  each trial is greater than 0 by some finite amount, regardless the history \(\mathcal {H}_{t-1}\) , the infinite numbers of allocation statement is true. As shown in Eq. (REF ), in the case of TS with Gaussian rewards, the allocating probabilities at each finite step \(t\)  are positive regardless the arm as they are given by the CDF of a normal distribution calculated in a finite value.
